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On the ASI Concept
Copyright © IFAC C ontrol Science and Technology (8th Triennial World Congress) Kyoto. Japan . 1981
ON THE ASI CONCEPT N. S. Rajbman and V. M. Chadeev Institute of Control Sciences, Moscow, USSR
Abstract. Adaptive systems of control with an identifier (ASI) are a new class of control systems for nonstationary plants and the plants whose transfer functions are not known a priori. It is shown that when the ASI is applied to a plant whose transfer function for the disturbance channels is not known and for the control channel is known, the identification does not deteriorate at closing of the loop. At present the ASI concepts have found application in designing of control s~stems for various industries-metallurgy, chemistry, oil chemistry, shipping etc. The paper gives a basic nonlinear equation of the ASI, considers basic modes of operation. Examples of application are also given. Keywords. Adaptive control; identification; time-varying systems; closed-loop systems, quality control.
INTRODUCTION Adaptive systems of control with an identifier (ASI) are a new class of control systems designed to control nonstationary plants and the plants whose transfer functions are not known a priori. The ASI's open up a possibility to standardize systems for control of complex plants of various physical nature. This paper deals with basic concepts of the ASr. Block-diagram of ASI A block-diagram for one plant is shown in Fig. 1, where X,Y,J and U are vectors of input, output, specified output and control variables, respectively. Specified output variables J are quantitative characteristics of control circuits specified by performance data, standards etc. and include both the technical and economic indices. Identification and control functions in the ASI can be implemented as individual technical means. The experience has shown, however, that one can gain more by their program separation in the control computer, one or several, depending on complexity of the plant controlled. Identification is generally carried out as two stages, strategical and tactical iden1099
tifiers. Both stages are used when apriori information is scarce or absent at all, and the mathematical model should be built using inputoutput data. The ASI has its core in identification process which determines the efficiency of the whole system. If the identification problem solution yielded a model optimal in some respect, the entire system can be optimized. ASI has two modes of operation: learning and control. Both are automatic. The learning is carried out in identific~tion units. After terminating the learning process the system switches automatically to a control mode. In that mode the current identification is automatically carried on updating the changing parameters. Strategical identifier in ASI solves a wider identification problem. It selects data variables and the optimality criterion, the type and tension of input output linkage, chooses the structure of the basic model and determines its parameters, estimates the degree of a model's identity to the plant, and the systems efficiency. The problem solution in the strategical identifier is maintained "off-line", beyond the loop and is associated with accumulation of data from the sensors and its subsequent processing. In the strategical identifier "classic"
1100
N. S. Rajbman and V. M. Chade e v
algorithms are applied to data processing, i.e. least-squares, maximum likelihood etc. Introduction of the strategical identifier into ASI is caused by a need to update regularly the listed and some other characteristics due to the plant modifications, emergence of new measuring means etc.
plant. The motion of the plant proper is described by the first equation in the system (1). Application of ASI is reasonable only when at least the parameter vector H is unknown in the plant equation. The second equation describes the controller, while the third is the equation of the identifier operation.
Tactical identifier of ASI is designed for current estimation of control system parameters. Parameter modifications during the plant's functioning are followed by the tactical identifier and the results are passed over to the control unit of ASI. The basic data for solving the identification problem here are the results obtained in the strategical identifier and the current data on the results of measuring input and output variables. Identification in the tactical identifier is done on-line, in real time. Recurrent algorithms are usually used here, they require the minimal memory volume, have fast response and are simple for implementation.
Note also that the ASI equation will be nonlinear even for linear G and \'1 of the plant and controller since the application of algorithm F to obtaining the vector of estimates K of the unknown parameters vector H can be only nonlinear. Hence, the adaptive system equation can be reduced to a nonlinear equattion of non-adaptive system. Substituting K from the third equation (1) into the second one we obtain
ASI controller determines control U which is necessary to obtain the given value of J • The control U is determined using the system's model obtained in the strategical identifier, the coefficients obtained in the tactical identifier, and J • Optimal control in this unit is found by using one of the two approaches: the extremum for the technicoeconomical index is determined with restrictions on quality J ,or the best quality indi ces of J are determined with restrictions on the technico-economical index. The basic equation of ASI The motion of the adaptive control system shown in Fig. 1 is described by the following system of equations y=G(H,X,Y,U) u=;/ ( K,X ,Y ,U) K=F(X,Y,U)
(1)
where Y is the plant output, X is the vector of plant inputs including the past history, Y is the vector of plant outputs including the past history, H is the vector of unknown parameters of the plant, u is the control vector, U is the vector of controls including the past history, K is the estimated vector of unknown parameters of the
u=W(F(X,Y,U),X,Y,U).
(2)
This equation includes only variables which can be measured explicitly. The equation also contains the plant equation since the plant output Y and input variables X,U are linked through operator G • To analyze the ASI behaviour in a general case is a challenge since the system (1) is nonlinear and plant parameters H in practically relevant problems are not stationary. BeSides, the analysis gets even more complicated due to inevitable noise. To be able to analyze the system's behaviour we have to make additional assumptions on properties of parameters and signals. ASI operation modes The identification quality is crucial for the ASI functioning. If a model fits the plant poorly, a poor control may result. Correspondence of a model to the plant for models linear in parameters should be determined from the inequality (3)
This criterion, however, can only be used in theoretical studies. In practice indirect criteria are made use of since the plant parameters H are not known. Specifically, the criterion of the plant and model output difference is widespread in practice r=M 11 y-y* 11 '
On the ASI Co nc ept
where y is the plant output, y" is the model output, 1,1 is mathematical expectation. Criteria () and (4) are linked through input vectors of the plant. For example, the plant operator A for plants linear in parameters can be expanded in the known operators A.
~
y = (H,A(X»=a Sum h. A. ~
~
~x.
~
)
(5)
where hi are unknown paramet ers of the plant, Ai are known operators ( • , .) are scalar product symbois. If the plant model is looked for within the same structure we yield the following for the model
y*=(K, A(X» ;l}iAi(X)
(6)
l-
where k i are parameter estimates for the plant hi Hence, criteria () and (4) as follows from (5) and (6) are related like
y-y*;«H-K,A(X». One can easily see that the small difference (4) is a condition necessary but insufficient of the smallness of the difference r from (). For convenience we can separate two modes of ASI functioning. First, that of learning, and second, that of control. The first mode has a large identification error inherent and is rarely met with, usually only during the system running-in. When the identification error r is great compared to the variation range of the unknown estimated parameters h , the analysis of identification dynamics can be simplified. In first approximation to estimate the duration of the transition period with great initial errors we can ignore the plant's nonstationarity. Numerous studies (Rajbman, Chadeev, 1980) show that at this stage of system functioning the high convergence rate is yielded by adaptive algorithms with little smoothing. Specifically, in a simplest one-step adaptive algorithm widely used
11 0 1
assumed zero. The system switches over to the control mode when identification errors are small. An ideal criterion for transition to the control mode would have been the smallness of estimates r • Due to impossibility to measure r in practice, however, the small enough degree of plant's uncertainty is used, i.e. the smallness of the ratio of the prediction error variance D(y-y*) to the output variance Dy • Transition from learning to control and back can be maintained manually and automatically. The decision-taking on switching over the modes is associated with some risk due to the lack of measurable criterion of identification quality. Transition to the control mode with the model, not accurate yet, may lead not only to the current losses in control but can also have the long run consequences. Introduction of control in some cases dealt with in detail later in the ' paper, hinders the identification and consequently takes more time before a satisfactory control can be gained. The control mode in ASI is rem arkable in that the identifier follows the sporadically changing plant parameters H • The identifier quality is assessed not by the duration of the transition process but by the average tracing error. ASI applications ASI concepts have found the greatest application in systems of process control. They make up for disturbances affecting the processes when the transfer function channels are not only unknown but are also changing. I~ is f~eld
important that in this primary of application there arise no problems of the plant identification (via disturbance channels) when the system closes up, i.e. when it transfers from the mode of learning to that of control. We will dwell a bit longer on that since it is of great significance in practice. The equation for a plant linear parameter-wise can be written as ; AY +BU +CX
y N
the posit i ve paramet er a at the initial stage of learning should be
N
N
(9) N
where YN is the scalar output of the plant at the moment N ; A is a row-vector of known plant parameters with a dimensionality M
N. S. Rajbman and V. M. Chade ev
110 2
is a column-vector of past inputs (Y n=(YlT-1 'YN- 2 "" 'YiJ-f,;»;B is a rowvector of known plant parameters via control channels with dimensionality s U is a column-vector of controls (U~T= (U1\T' u:r 1"'" U.. 1. 1', ,N-s+1 )) c is a row-vector of unknown plant parameters via disturbance channels with dimensionality II ; X is a columnvector of input distur~ances with dimensionality n T
The equation of the plant's model (10)
improving and this improvement is not a fun ction of control which is important. Hence, identification in both open and closed loops is carried out similarly. Assuming that the inputs are independent we can get the convergence rate hl(C-C*)( C -C*)~(1-j)N ( C-C*)(C-C*)t N N ~ 0 0 where ion.
~I
is mathematical expectat-
The positive parameter a in the algorithm (11) improves considerably the identifier chara c teristics with the noise present.
differs from that of the plant in that the vector of unknown parameters C is substituted by the vector of estimates
If vectors A and B in equation (9) are also known with a certain error ( .! A and A B ) , the prediction error will depend on the controls and the past outputs, i.e.
We will use a one-step algorithm (8) for plant identification and the subsequent updating of the parameters. It can be written as follows:
Ev¥d~ntlY tNe alggrithm applfcation
Note, and this is also relevant for subsequent speculations, that only input disturbances XlI are used as input vectors (in a general case a common vector including disturbances, controls and past inputs can be used as an input vector). A simple check shows that the prediction error (y-y*) depends only on the disturbance vector X and does not depend on controls and past outputs. Indeed, it directly follows from (9) and (10) that YN-YN =(C-C*)~. (12) Let us calculate the identification error when using (11), for a=O We yield
(C-C~ )(C-CI)t=(C-CI_1 ) (C-CI_1 )! -(cx -C* N
N-1
X)/xt x 1I
N 1I
(13)
Using the Cauchy-Bunyakovsky's inequality we get (C-C* ) (C-C* ) t~( C-C* ) (C-C*) t(14) N N N-1 N-1 i.e. the estimates of the unknown plant parameters are constantly
y-y*
=~A'Y +~B'U
+ (C-C*) X •
(11) will cause here an additional error of identification, which even asymptotically in a general case will not converge to a point. It should be emphasized, however that a systematic component of the prediction error due to ~ A and A B will be compensated fully by the identifier, and only a random component will result in generating a certain convergence area. At present ASI concepts are widely applied to designing of control systems in various industries. The greatest experience has been accumulated in metallurgy where the systems have been operational since 1973. Control systems using the identifier for restoration of plant characteristics have been implemented in chemistry, shipping (Rajbman, Chadeev, 1980; Danilov, Rajbman and others, 1980; Astrom, Kallstrom, 1976). The ASI's used in industry gain an economic effect primarily due to reduction in the variance of the basic quality index. REFERENCES Rajbman, N.S., and V.M. Chadeev (1980). Identification of industrial processes. North-Holland, Amsterdam. Danilov, F.A., Rajbman N.S. and others. (1980). Adaptive control of rolling accuracy. Metallur~iya. M. Astrom K.J., C.G.Kallstrom (1976». Identification of Ship Steering Dynamics. Automatica. Pergamon Press, V.12, No.1, January.
1103
On the ASI Concept
X
,.I
y ..
Plant U
I Controller
TK
r-1" Tactical
~ Identifier
Y
..
~
J
u I'
T Strategical Id.entifier I'
Fig. 1. Block-diagram of ASI.
Discussion to Paper 36.6 M.J. Grimble (Scotland): Could you explain the marine application you mentioned in your lecture? What type of vessel is this control method applied to ? V. Lototsky (USSR): The AS! implementation to hot tube rolling mill control includes the minicomputer, which carries out the functions of identifier and controller, several sensors for measuring input control, and output variables, and relevant software, which includes algorithms of identification and control, as well as security, monitoring and other additional functions. As for implementation of ASI in shipping, I am less informed on the issue, so I'll ask Dr. Chadeer to answer in writing. The on-line operational mode is needed due to rather fast changes in rawmaterials parameters. At the same time the slow changes in the plant parameters can be handled in off-line mode.
ON THE ASI CONCEPT N. S. Rajbman and V. M. Chadeev Institute of Control Sciences, Moscow, USSR
Abstract. Adaptive systems of control with an identifier (ASI) are a new class of control systems for nonstationary plants and the plants whose transfer functions are not known a priori. It is shown that when the ASI is applied to a plant whose transfer function for the disturbance channels is not known and for the control channel is known, the identification does not deteriorate at closing of the loop. At present the ASI concepts have found application in designing of control s~stems for various industries-metallurgy, chemistry, oil chemistry, shipping etc. The paper gives a basic nonlinear equation of the ASI, considers basic modes of operation. Examples of application are also given. Keywords. Adaptive control; identification; time-varying systems; closed-loop systems, quality control.
INTRODUCTION Adaptive systems of control with an identifier (ASI) are a new class of control systems designed to control nonstationary plants and the plants whose transfer functions are not known a priori. The ASI's open up a possibility to standardize systems for control of complex plants of various physical nature. This paper deals with basic concepts of the ASr. Block-diagram of ASI A block-diagram for one plant is shown in Fig. 1, where X,Y,J and U are vectors of input, output, specified output and control variables, respectively. Specified output variables J are quantitative characteristics of control circuits specified by performance data, standards etc. and include both the technical and economic indices. Identification and control functions in the ASI can be implemented as individual technical means. The experience has shown, however, that one can gain more by their program separation in the control computer, one or several, depending on complexity of the plant controlled. Identification is generally carried out as two stages, strategical and tactical iden1099
tifiers. Both stages are used when apriori information is scarce or absent at all, and the mathematical model should be built using inputoutput data. The ASI has its core in identification process which determines the efficiency of the whole system. If the identification problem solution yielded a model optimal in some respect, the entire system can be optimized. ASI has two modes of operation: learning and control. Both are automatic. The learning is carried out in identific~tion units. After terminating the learning process the system switches automatically to a control mode. In that mode the current identification is automatically carried on updating the changing parameters. Strategical identifier in ASI solves a wider identification problem. It selects data variables and the optimality criterion, the type and tension of input output linkage, chooses the structure of the basic model and determines its parameters, estimates the degree of a model's identity to the plant, and the systems efficiency. The problem solution in the strategical identifier is maintained "off-line", beyond the loop and is associated with accumulation of data from the sensors and its subsequent processing. In the strategical identifier "classic"
1100
N. S. Rajbman and V. M. Chade e v
algorithms are applied to data processing, i.e. least-squares, maximum likelihood etc. Introduction of the strategical identifier into ASI is caused by a need to update regularly the listed and some other characteristics due to the plant modifications, emergence of new measuring means etc.
plant. The motion of the plant proper is described by the first equation in the system (1). Application of ASI is reasonable only when at least the parameter vector H is unknown in the plant equation. The second equation describes the controller, while the third is the equation of the identifier operation.
Tactical identifier of ASI is designed for current estimation of control system parameters. Parameter modifications during the plant's functioning are followed by the tactical identifier and the results are passed over to the control unit of ASI. The basic data for solving the identification problem here are the results obtained in the strategical identifier and the current data on the results of measuring input and output variables. Identification in the tactical identifier is done on-line, in real time. Recurrent algorithms are usually used here, they require the minimal memory volume, have fast response and are simple for implementation.
Note also that the ASI equation will be nonlinear even for linear G and \'1 of the plant and controller since the application of algorithm F to obtaining the vector of estimates K of the unknown parameters vector H can be only nonlinear. Hence, the adaptive system equation can be reduced to a nonlinear equattion of non-adaptive system. Substituting K from the third equation (1) into the second one we obtain
ASI controller determines control U which is necessary to obtain the given value of J • The control U is determined using the system's model obtained in the strategical identifier, the coefficients obtained in the tactical identifier, and J • Optimal control in this unit is found by using one of the two approaches: the extremum for the technicoeconomical index is determined with restrictions on quality J ,or the best quality indi ces of J are determined with restrictions on the technico-economical index. The basic equation of ASI The motion of the adaptive control system shown in Fig. 1 is described by the following system of equations y=G(H,X,Y,U) u=;/ ( K,X ,Y ,U) K=F(X,Y,U)
(1)
where Y is the plant output, X is the vector of plant inputs including the past history, Y is the vector of plant outputs including the past history, H is the vector of unknown parameters of the plant, u is the control vector, U is the vector of controls including the past history, K is the estimated vector of unknown parameters of the
u=W(F(X,Y,U),X,Y,U).
(2)
This equation includes only variables which can be measured explicitly. The equation also contains the plant equation since the plant output Y and input variables X,U are linked through operator G • To analyze the ASI behaviour in a general case is a challenge since the system (1) is nonlinear and plant parameters H in practically relevant problems are not stationary. BeSides, the analysis gets even more complicated due to inevitable noise. To be able to analyze the system's behaviour we have to make additional assumptions on properties of parameters and signals. ASI operation modes The identification quality is crucial for the ASI functioning. If a model fits the plant poorly, a poor control may result. Correspondence of a model to the plant for models linear in parameters should be determined from the inequality (3)
This criterion, however, can only be used in theoretical studies. In practice indirect criteria are made use of since the plant parameters H are not known. Specifically, the criterion of the plant and model output difference is widespread in practice r=M 11 y-y* 11 '
On the ASI Co nc ept
where y is the plant output, y" is the model output, 1,1 is mathematical expectation. Criteria () and (4) are linked through input vectors of the plant. For example, the plant operator A for plants linear in parameters can be expanded in the known operators A.
~
y = (H,A(X»=a Sum h. A. ~
~
~x.
~
)
(5)
where hi are unknown paramet ers of the plant, Ai are known operators ( • , .) are scalar product symbois. If the plant model is looked for within the same structure we yield the following for the model
y*=(K, A(X» ;l}iAi(X)
(6)
l-
where k i are parameter estimates for the plant hi Hence, criteria () and (4) as follows from (5) and (6) are related like
y-y*;«H-K,A(X». One can easily see that the small difference (4) is a condition necessary but insufficient of the smallness of the difference r from (). For convenience we can separate two modes of ASI functioning. First, that of learning, and second, that of control. The first mode has a large identification error inherent and is rarely met with, usually only during the system running-in. When the identification error r is great compared to the variation range of the unknown estimated parameters h , the analysis of identification dynamics can be simplified. In first approximation to estimate the duration of the transition period with great initial errors we can ignore the plant's nonstationarity. Numerous studies (Rajbman, Chadeev, 1980) show that at this stage of system functioning the high convergence rate is yielded by adaptive algorithms with little smoothing. Specifically, in a simplest one-step adaptive algorithm widely used
11 0 1
assumed zero. The system switches over to the control mode when identification errors are small. An ideal criterion for transition to the control mode would have been the smallness of estimates r • Due to impossibility to measure r in practice, however, the small enough degree of plant's uncertainty is used, i.e. the smallness of the ratio of the prediction error variance D(y-y*) to the output variance Dy • Transition from learning to control and back can be maintained manually and automatically. The decision-taking on switching over the modes is associated with some risk due to the lack of measurable criterion of identification quality. Transition to the control mode with the model, not accurate yet, may lead not only to the current losses in control but can also have the long run consequences. Introduction of control in some cases dealt with in detail later in the ' paper, hinders the identification and consequently takes more time before a satisfactory control can be gained. The control mode in ASI is rem arkable in that the identifier follows the sporadically changing plant parameters H • The identifier quality is assessed not by the duration of the transition process but by the average tracing error. ASI applications ASI concepts have found the greatest application in systems of process control. They make up for disturbances affecting the processes when the transfer function channels are not only unknown but are also changing. I~ is f~eld
important that in this primary of application there arise no problems of the plant identification (via disturbance channels) when the system closes up, i.e. when it transfers from the mode of learning to that of control. We will dwell a bit longer on that since it is of great significance in practice. The equation for a plant linear parameter-wise can be written as ; AY +BU +CX
y N
the posit i ve paramet er a at the initial stage of learning should be
N
N
(9) N
where YN is the scalar output of the plant at the moment N ; A is a row-vector of known plant parameters with a dimensionality M
N. S. Rajbman and V. M. Chade ev
110 2
is a column-vector of past inputs (Y n=(YlT-1 'YN- 2 "" 'YiJ-f,;»;B is a rowvector of known plant parameters via control channels with dimensionality s U is a column-vector of controls (U~T= (U1\T' u:r 1"'" U.. 1. 1', ,N-s+1 )) c is a row-vector of unknown plant parameters via disturbance channels with dimensionality II ; X is a columnvector of input distur~ances with dimensionality n T
The equation of the plant's model (10)
improving and this improvement is not a fun ction of control which is important. Hence, identification in both open and closed loops is carried out similarly. Assuming that the inputs are independent we can get the convergence rate hl(C-C*)( C -C*)~(1-j)N ( C-C*)(C-C*)t N N ~ 0 0 where ion.
~I
is mathematical expectat-
The positive parameter a in the algorithm (11) improves considerably the identifier chara c teristics with the noise present.
differs from that of the plant in that the vector of unknown parameters C is substituted by the vector of estimates
If vectors A and B in equation (9) are also known with a certain error ( .! A and A B ) , the prediction error will depend on the controls and the past outputs, i.e.
We will use a one-step algorithm (8) for plant identification and the subsequent updating of the parameters. It can be written as follows:
Ev¥d~ntlY tNe alggrithm applfcation
Note, and this is also relevant for subsequent speculations, that only input disturbances XlI are used as input vectors (in a general case a common vector including disturbances, controls and past inputs can be used as an input vector). A simple check shows that the prediction error (y-y*) depends only on the disturbance vector X and does not depend on controls and past outputs. Indeed, it directly follows from (9) and (10) that YN-YN =(C-C*)~. (12) Let us calculate the identification error when using (11), for a=O We yield
(C-C~ )(C-CI)t=(C-CI_1 ) (C-CI_1 )! -(cx -C* N
N-1
X)/xt x 1I
N 1I
(13)
Using the Cauchy-Bunyakovsky's inequality we get (C-C* ) (C-C* ) t~( C-C* ) (C-C*) t(14) N N N-1 N-1 i.e. the estimates of the unknown plant parameters are constantly
y-y*
=~A'Y +~B'U
+ (C-C*) X •
(11) will cause here an additional error of identification, which even asymptotically in a general case will not converge to a point. It should be emphasized, however that a systematic component of the prediction error due to ~ A and A B will be compensated fully by the identifier, and only a random component will result in generating a certain convergence area. At present ASI concepts are widely applied to designing of control systems in various industries. The greatest experience has been accumulated in metallurgy where the systems have been operational since 1973. Control systems using the identifier for restoration of plant characteristics have been implemented in chemistry, shipping (Rajbman, Chadeev, 1980; Danilov, Rajbman and others, 1980; Astrom, Kallstrom, 1976). The ASI's used in industry gain an economic effect primarily due to reduction in the variance of the basic quality index. REFERENCES Rajbman, N.S., and V.M. Chadeev (1980). Identification of industrial processes. North-Holland, Amsterdam. Danilov, F.A., Rajbman N.S. and others. (1980). Adaptive control of rolling accuracy. Metallur~iya. M. Astrom K.J., C.G.Kallstrom (1976». Identification of Ship Steering Dynamics. Automatica. Pergamon Press, V.12, No.1, January.
1103
On the ASI Concept
X
,.I
y ..
Plant U
I Controller
TK
r-1" Tactical
~ Identifier
Y
..
~
J
u I'
T Strategical Id.entifier I'
Fig. 1. Block-diagram of ASI.
Discussion to Paper 36.6 M.J. Grimble (Scotland): Could you explain the marine application you mentioned in your lecture? What type of vessel is this control method applied to ? V. Lototsky (USSR): The AS! implementation to hot tube rolling mill control includes the minicomputer, which carries out the functions of identifier and controller, several sensors for measuring input control, and output variables, and relevant software, which includes algorithms of identification and control, as well as security, monitoring and other additional functions. As for implementation of ASI in shipping, I am less informed on the issue, so I'll ask Dr. Chadeer to answer in writing. The on-line operational mode is needed due to rather fast changes in rawmaterials parameters. At the same time the slow changes in the plant parameters can be handled in off-line mode.